"Every possible history starting from a particular state and ending at a particular state is assigned a complex number by some predefined rules in particular that the complex number is the product of the complex numbers assigned to each part of the history.
The complex numbers are summed and the square magnitude gives the probability for the final state given the initial state."
This is my very brief summary of the essence of quantum mechanics. The problem is that it necessitates the definition of the sum and product of complex numbers. Feynman used arrows instead in some public lectures. Do you know of any better explanations that don't involve complex numbers or little arrows?
Also, the definition is not entirely true as you have to regulate the sums or do Wick rotation tricks otherwise you just get infinity. Which sort of suggests that this definition might not be the best.